# A fun computer aided number theory project for kids

Not surprisingly, today’s Family Math began with a neat question that James Tanton posted on Twitter:

I chose to not go through the entire problem but thought it would be fun to answer a slightly easier question – what is the smallest positive integer with exactly 100 divisors?

We started by talking through the problem and looking for some ideas on where to start. Turned out that each kid had an interesting idea, and I thought it would be fun to pursue both of them:

We started off down the path suggested by my younger son – start by looking to see if there are any patterns in the number of divisors on the integers. The difficulty is that although there is a pattern, in fact a really cook pattern,  it is a little tough to see. I let them noodle on it for a while and then asked them to talk a little bit about what it means to be a divisor of a number.  That led to my older son noticing a similarity in the prime factors.

Next we moved to looking at the factors of a larger number -> 24.  After understanding the divisors of 24, we built off of the counting project we’ve been doing this summer to see how to find an easier way count the number of factors from the primes. Making this connection was the reason I wanted to do this little project.

My younger son’s idea led us to the rule for counting the number of divisors of of a given integer. Next we move on to my older son’s idea of looking for a pattern when we list out the smallest integers with a given number of factors. To aide in this part of the project we used Mathematica and had a lot of fun looking at lots of patterns in the numbers (also sorry about the camera issue in this video):

Having checked what the smallest integer was that had N factors for N ranging from 1 to 10, we went back to the whiteboard to look more carefully at those numbers. Here my lack of planning ahead came back to bite me a little since the prime factors of all of these numbers had only 2’s and 3’s. To correct for that I added in the smallest number with exactly 12 factors – that integer is 60.

Talking through the pattern we see in this list lets us take a couple of guesses at the smallest integer with exactly 100 factors. Looking at our guesses, the smallest one is $2^4 * 3^4 * 5^1 * 7^1.$

Having gone through the difficult part of the problem, we head back to Mathematica to see if our guess is correct. Mathematica gives us a short list of numbers with exactly 100 factors, and our number is indeed the smallest!

Finally, we wrap up by taking a quick look at the second smallest number with 100 factors. When we factor it into primes we see that it looks pretty similar to the number we found – just one prime factor is different.

When you see the original problem for the first time it seems almost impossible to solve. A little bit playing around with patterns leads to the amazing discovery that the problem isn’t as intractable as it seems, though. You also get a look at several really cool patterns relating integers to their prime factors. Definitely a fun project and a fun way to show how computers can be helpful in solving problems, too.

# Working through some NY State test questions (3/3) 8th Grade

[this is the 3rd in a 3 part series. I’ll use the same introduction for each. The first post is here:

and the second post is here:

Note on this third post – I wrote the post all the way through and then geniusly deleted it rather than publishing it. Sorry if the rewritten post reads like I wrote it too fast. I did.

]

I saw this interesting link posted by a couple of people in the last few days:

How Would You Score On A Third-Grade Common Core Math Test?

To say the least, there’s been a lot written about the Common Core standards and their impact on education in the US. I haven’t followed this debate carefully, nor have I learned much about the standards. With the release of these questions I thought it would be interesting to get a glimpse of testing done in one state. So for today’s Family Math project I asked each kid to work through each of the 15 questions. This post is about the last 5 questions – the 5 from the 8th grade exam. My younger son will be going into 3rd grade this year and my older son will be going into 5th.

Question 1: What is the solution to the following system of equations?  3x + 4y = -2, and 2x – 4y = -8.

For me this isn’t a great multiple choice question, and my younger son shows why.  He is able to answer the question by just plugging the answers back into the equation. No real understanding necessary.

My older son has seen a little bit about solving equations with more than one variable and does recall a pretty good way to solve this problem.

Question 2: Which phrase describes a nonlinear function?

I’ve not covered functions with my younger son, so he does not understand the question. I thought the question was interesting enough to talk through with him, though, just as a very basic (and quick) introduction to linear functions.

My older son recognizes that the first choice – the area of a circle in terms of the radius – is nonlinear. That enables him to recognize that as the correct choice. However, he does not understand the functions described in choices (c) and (d), so we spend a few minutes talking through the functions described in these two choices.

Question 3:  Which number is equivalent to $3^4 / 3^2$

Both kids have seen the power notation before, so I do not expect this one to be confusing.  My younger son multiplies out the top and bottom of the fraction and then divides.  I ask him if there is any other way to approach this problem, and he mentions subtracting the powers.

My older son takes an interesting approach and actually writes out the powers as repeated multiplication. He then cancels two threes from the top and bottom leaving 9. When I ask him if there is any other way to approach the problem he explains why the remaining choices cannot be correct.

Question 4: Determine the product. 800.5 x (2 x 10^6)

I haven’t really covered scientific notation with my younger son, so I expected this to be confusing to him. He did have a couple of interesting ideas, but this problem was more of a discussion between the two of us than a solution to the problem.

My older son has seen a little bit of scientific notation and does manage to reason out the solution in a way that was pretty different that what I was expecting.

Question 5: Which expression is equivalent to 4^7 x 4^-5?

This one is so similar to question 3 that I wonder why the Huffington post chose it for their article. I decided to probe a little deeper into their understanding of powers after they answered the question.

My younger son had a little trouble explaining the rule for multiplying powers. It turned out that he was struggling to find a name for the rule, but when I asked him to explain the rule in words he got pretty close. That was nice.

My older son understood that 4^(-5) was the same as 1 / 4^5 and solved the problem from there. I asked him where the other choices came from and he was able to come up with good answers.

Well, 30 problems in a couple of hours was tough, but we got it done. My younger son was able to understand 13 of the 15 questions and my older son, I think, understood all of them. As I mentioned in part 2 of this series, I was a little surprised by the lack of depth in the problems, though the sample size here is pretty small. Really only the question from the 8th grade test about linear functions had any depth to it. The rest could be answered by memorizing rules, or simply checking the answers.

For all the talk and controversy about the Common Core, honestly the questions don’t look that different from the SAT, ACT, and CAT questions I remember from the 80s.